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flop : i think its fine to bet since we can get value from flop and turn if we hit our K or flush. i could make it bigger though for more fold equity like 4-4.3. whats his VPIP like? and fold to CB? for all you know if hes wide enough betting K9s here could be for value like against 78 89 79 and worse backdoor straight hands like 8J 9Q type of hand if hes actually wide enough.

turn: we now have a GS to go with our FD together with an overcard K. what is villian VPIP, fold to CB on flop and turn like? and his bet vs missed CB? if he has a moderate to high fold to CB on turn and VPIP i agree with the bet because some hands that called could flop could fold here. like 9Q 8J maybe even AJ KQ overcards or weak pairs like A2 A6 etc. and even if called we still have the overcard K against them together with our FD and GS. perhaps we can also go bigger on our bet to maximise fold equity and set up a possible shove on the river.

river: what is villain range on the river? if he has low fold to CB on flop and turn it means he carries a wide range of draws and weak hands to the river. does villain fold a hand like 88 here? how about a hand like AT? i think his WTS lends us some insight. if its like >30% here i think hes not folding even 88 here so we might want to check and give up. but if its like >15% here and he has low fold to CB on flop and turn suggesting a wide range of draws what could fold that went to the river i agree with a bluffbet but i think your bet is too small to fold out a hand like T, probably need to go bigger like 20-22 for close to full pot. that is what you would have bet if you hit the A right?

On that middle card flop and the turn which gave more draws and made it wetter I don't think he's folding this turn. However if we check I think he's going to bet turn so what's our plan? I could think of a check raise semi bluff and jamming on over cards like JQTKA as a bluff since his turn betting range is weak.

We could also check call to draw and possibly be ahead sometimes with K high. I think betting is fine too he could fold some random hands he called with on the flop.